Quotes about algebra

A collection of quotes on the topic of algebra, use, mathematics, science.

Quotes about algebra

“I obtained scientific guidance and stimulation mainly through personal mathematical contacts in Erlangen and in Göttingen. Above all I am indebted to Mr. E. Fischer from whom I received the decisive impulse to study abstract algebra from an arithmetical viewpoint, and this remained the governing idea for all my later work.”

Emmy Noether (1882–1935) German mathematician

Mathematics, Idea, Decision, Study

“Any organism must be treated as-a-whole; in other words, that an organism is not an algebraic sum, a linear function of its elements, but always more than that.”

Alfred Korzybski (1879–1950) Polish scientist and philosopher

Source: Science and Sanity (1933), p. 64.
Context: Any organism must be treated as-a-whole; in other words, that an organism is not an algebraic sum, a linear function of its elements, but always more than that. It is seemingly little realized, at present, that this simple and innocent-looking statement involves a full structural revision of our language...

“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.”

Omar Khayyám (1048–1131) Persian poet, philosopher, mathematician, and astronomer

Treatise on Demonstration of Problems of Algebra (1070).

Life, People, Truth, Fools

“As a graduate student I studied mathematics fairly broadly and I was fortunate enough, besides developing the idea which led to "Non-Cooperative Games," also to make a nice discovery relating to manifolds and real algebraic varieties. So I was prepared actually for the possibility that the game theory work would not be regarded as acceptable as a thesis in the mathematics department and then that I could realize the objective of a Ph. D. thesis with the other results.”

John Nash (1928–2015) American mathematician and Nobel Prize laureate

Autobiographical essay (1994)

Game, Mathematics, Idea, Study

“The elements of justice are identical with those of algebra.”

Pierre Joseph Proudhon (1809–1865) French politician, mutualist philosopher, economist, and socialist

Source: What is Property? (1840), Ch. IV

Justice

“When I began the following Work, my only object was to extend and explain more fully the Memoir which I read at the public meeting of the Academy of Sciences in the month of April 1787, on the necessity of reforming and completing the Nomenclature of Chemistry. While engaged in this employment, I perceived, better than I had ever done before, the justice of the following maxims of the Abbé de Condillac, in his System of Logic, and some other of his works. "We think only through the medium of words.—Languages are true analytical methods.—Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method.—The art of reasoning is nothing more than a language well arranged."”

Antoine Lavoisier (1743–1794) French chemist

Source: Elements of Chemistry (1790), p.xiii

Justice, Art, Thinking, Science

“We live through myriads of seconds, but there is only one second among all these myriads which brings our whole inner world to the boil; the second in which, as Stendhal described, there suddenly takes place a crystallization in the supersaturated blood; a magical second like that of procreation, and, like it, hidden in the warm interior of one's own body, invisible, intangible, impalpable, a unique experience of mystery. No algebra of the soul can calculate it; no alchemy can divine it. Usually, even for ourselves, it remains unsearchable.”

Stefan Zweig (1881–1942) Austrian writer

Confusion of Feelings or Confusion: The Private Papers of Privy Councillor R. Von D (1927)

World, Soul

“As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme,Tr. McCormack, cited in Robert Edouard Moritz, Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. V The teaching of mathematics, p. 81. https://archive.org/stream/memorabiliamathe00moriiala#page/80/mode/2up

Science, Perfection

“Through algebra you easily arrive at equations, but always to pass therefrom to the elegant constructions and demonstrations which usually result by means of the method of porisms is not so easy, nor is one's ingenuity and power of invention so greatly exercised and refined in this analysis.”

Isaac Newton (1643–1727) British physicist and mathematician and founder of modern classical physics

The Mathematical Papers of Isaac Newton (edited by Whiteside), Volume 7; Volumes 1691-1695 / pg. 261. http://books.google.com.br/books?id=YDEP1XgmknEC&printsec=frontcover Geometriae (Treatise on Geometry)

Exercises

“Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Riemann, the more geometric approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century "American" school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods.”

Robin Hartshorne book Algebraic Geometry

Algebraic Geometry, Springer, (1977), p. xiii
Algebraic Geometry (1977)

School, Problem

“Whereas in Arithmetick Questions are only resolv'd by proceeding from given Quantities to the Quantities sought, Algebra proceeds in a retrograde Order, from the Quantities sought as if they were given, to the Quantities given as if they were sought, to the End that we may some Way or other come to a Conclusion or Æquation, from which one may bring out the Quantity sought.”

Isaac Newton book Arithmetica Universalis

Arithmetica Universalis (1707)
Context: Whereas in Arithmetick Questions are only resolv'd by proceeding from given Quantities to the Quantities sought, Algebra proceeds in a retrograde Order, from the Quantities sought as if they were given, to the Quantities given as if they were sought, to the End that we may some Way or other come to a Conclusion or Æquation, from which one may bring out the Quantity sought. And after this Way the most difficult problems are resolv'd, the Resolutions whereof would be sought in vain from only common Arithmetick. Yet Arithmetick in all its Operations is so subservient to Algebra, as that they seem both but to make one perfect Science of Computing; and therefore I will explain them both together.<!--pp.1-2

Way, Question

“Problem solving, and I don't mean algebra, seems to be my life's work. Maybe it's everyone's life's work.”

Beverly Cleary (1916) American writer of children's books

Life, Problem

“Friendship is an Algebra test that nobody passes.”

Gregory David Roberts book Shantaram

Source: Shantaram

Friendship

“… A strange art – music – the most poetic and precise of all the arts, vague as a dream and precise as algebra.”

Guy De Maupassant (1850–1893) French writer

Source: Complete Works

Dreams, Art, Music

“I write emotional algebra.”

Anaïs Nin (1903–1977) writer of novels, short stories, and erotica

“We have already various treatises on Mechanics, but the plan of this one is entirely new. I intend to reduce the theory of this Science, and the art of solving problems relating to it, to general formulae, the simple development of which provides all the equations necessary for the solution of each problem. I hope that the manner in which I have tried to attain this object will leave nothing to be desired. No diagrams will be found in this work. The methods that I explain require neither geometrical, nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.”

Joseph Louis Lagrange book Mécanique analytique

Mécanique analytique (1788) as quoted by E. W. Hobson, Mathematics, from the points of view of the Mathematician and of the Physicist (1912) an address delivered to the Mathematical and Physical Society of University College London, p.13. https://books.google.com/books?id=H7Y_AQAAIAAJ&pg=PA13

Love, Pleasure, Hope, Art

“Some other Course therefore must be taken to promote the Search of Knowledge. Some other kind of Art for Inquiry than what hath been hitherto made use of, must be discovered; the Intellect is not to he suffer'd to act without its Helps, but is continually to be assisted by some Method or Engine, which shall be as a Guide to regulate its Actions, so as that it shall not be able to act amiss: Of this Engine, no Man except the incomparable Verulam hath had any Thoughts, and he indeed hath promoted it to a very good pitch; but there is yet somewhat more to be added, which he seem'd to want time to compleat. By this, as by that Art of Algebra in Geometry, 'twill be very easy to proceed in any Natural Inquiry, regularly and certainly: And indeed it may not improperly be call'd a Philosophical Algebra, or an Art of directing the Mind in the search after Philosophical Truths, for as 'tis very hard for the most acute Wit to find out any difficult Problem in Geometry. without the help of Algebra to direct and regulate the Acts of the Reason in the Process from the question to the quœsitum, and altogether as easy for the meanest Capacity acting by that Method to compleat and perfect it, so will it be in the inquiry after Natural Knowledge.”

Robert Hooke (1635–1703) English natural philosopher, architect and polymath

"The Present State of Natural Philosophy, and wherein it is deficient," The Posthumous Works of Robert Hooke https://books.google.com/books?id=6xVTAAAAcAAJ (1705) ed., Richard Waller, pp. 6-7.

Truth, Art, Search, Question

“The truth is that Mozart, Pascal, Boolean Algebra, Shakespeare, parliamentary government, baroque churches, Newton, the emancipation of women, Kant, Marx, and Balanchine ballets don't redeem what this particular civilization has wrought upon the world. The white race is the cancer of human history.”

Susan Sontag (1933–2004) American writer and filmmaker, professor, and activist

Partisan Review (Winter 1967), p. 57

Women, Truth, World, History

“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.”

Michael Atiyah (1929–2019) British mathematician

[Michael Atiyah, Collected works. Vol. 6, The Clarendon Press Oxford University Press, Oxford Science Publications, http://www.math.tamu.edu/~rojas/atiyah20thcentury.pdf, 978-0-19-853099-2, 2160826, 2004]

Soul, Question

“THEY who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1840s, The Mathematical Analysis of Logic, 1847, p. ii: Lead paragraph of the Introduction

Truth, Law, Question, Problem

“If there were only some mechanism (like Seurat's proposed system of painting, or the projected Universal Algebra that Gödel believes Leibnitz to have perfected and mislaid) for reasonably and systematically converting into poetry what we see and feel and are! When one reads the verse of people who cannot write poems — people who sometimes have more intelligence, sensibility, and moral discrimination than most of the poets — it is hard not to regard the Muse as a sort of fairy godmother who says to the poet, after her colleagues have showered on him the most disconcerting and ambiguous gifts, "Well, never mind. You're still the only one that can write poetry."”

Randall Jarrell (1914–1965) poet, critic, novelist, essayist

"Verse Chronicle," The Nation (23 February 1946); reprinted as "Bad Poets" in Poetry and the Age (1953)
General sources

People, Feeling, Intelligence, Universe

“Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence.”

James Joseph Sylvester (1814–1897) English mathematician

James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.

Law, Mathematics, Idea, Effort

“This work… was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
A Treatise on Algebra (1842)

Light

“Lety5 - ay4 + by3 - cy2 + dy - e = 0be the general equation of the fifth degree and suppose that it can be solved algebraically,—i. e., that y can be expressed as a function of the quantities a, b, c, d, and e, composed of radicals. In this case, it is clear that y can be written in the formy = p + p1R1/m + p2R2/m +…+ pm-1R(m-1)/m,m being a prime number, and R, p, p1, p2, etc. being functions of the same form as y. We can continue in this way until we reach rational functions of a, b, c, d, and e. [Note: main body of proof is excluded]
…we can find y expressed as a rational function of Z, a, b, c, d, and e. Now such a function can always be reduced to the formy = P + R1/5 + P2R2/5 + P3R3/5 + P4R4/5, where P, R, P2, P3, and P4 are functions or the form p + p1S1/2, where p, p1 and S are rational functions of a, b, c, d, and e. From this value of y we obtainR1/5 = 1/5(y1 + α4y2 + α3y3 + α2y4 + α y5) = (p + p1S1/2)1/5,whereα4 + α3 + α2 + α + 1 = 0.Now the first member has 120 different values, while the second member has only 10; hence y can not have the form that we have found: but we have proved that y must necessarily have this form, if the proposed equation can be solved: hence we conclude that
It is impossible to solve the general equation of the fifth degree in terms of radicals.
It follows immediately from this theorem, that it is also impossible to solve the general equations of degrees higher than the fifth, in terms of radicals.”

Niels Henrik Abel (1802–1829) Norwegian mathematician

A Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree (1824) Tr. W. H. Langdon, as quote in A Source Book in Mathematics (1929) ed. David Eugene Smith

Way

“Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way"…. This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics.”

John C. Baez (1961) American mathematician and mathematical physicist

Way, Nature, Problem, Light

“In fifty words: Granted mobility, security (in the form of denying targets to the enemy), time, and doctrine (the idea to convert every subject to friendliness), victory will rest with the insurgents, for the algebraical factors are in the end decisive, and against them perfections of means and spirit struggle quite in vain.”

T. E. Lawrence (1888–1935) British archaeologist, military officer, and diplomat

The Evolution of A Revolt (1920)

Idea, Decision, Time, Victory

“The use of canon raised numerous questions concerning the paths of projectiles. …One might determine… what type of curve a projectile follows and…. prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 148

Question

“[T]he application of algebra to geometry… far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.”

John Stuart Mill (1806–1873) British philosopher and political economist

An Examination of Sir William Hamilton's Philosophy (1865) as quoted in 5th ed. (1878) p. 617. https://books.google.com/books?id=ojQNAQAAMAAJ&pg=PA617

Science

“There is only one other survey, Datta and Singh’s 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics.”

David Mumford (1937) American mathematician

[David Mumford, Book Review, Notices of the AMS, March 2010, 57, 3, http://www.ams.org/notices/201003/rtx100300385p.pdf]

History, Mathematics

“From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Music, Mathematics, Study

“I have endeavoured… to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are considered, and this difficulty has not been a little increased, in the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained.”

George Peacock (1791–1858) Scottish mathematician

Vol. II: On Symbolical Algebra and its Applications to the Geometry of Position (1845) Preface, p. iii
A Treatise on Algebra (1842)

Departure, Time

“The notions she fought with needed more than the simple algebra she’d been grudgingly taught at Lamai Hold. More and more she resented how they had robbed her of this, arguably her one talent, driving her from math and other abstractions by the simple expedient of making them seem boring.”

David Brin book Glory Season

Source: Glory Season (1993), Chapter 19 (p. 337)

“Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, p. 392

“When an equation…clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. …Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 143.

Time

“It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 48.
Theory of Numbers, 1892

“Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics.”

Florian Cajori (1859–1930) American mathematician

F. Cajori's Teaching and History of Mathematics in the U. S. (Washington, 1890), p. 265; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/198/mode/2up, (1914) p. 171; Persons and anecdotes.

Universe

“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.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Preface, p. iii
The Differential and Integral Calculus (1836)

Books, Mathematics, Idea, Knowledge

“What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze.”

Paul Klee (1879–1940) German Swiss painter

1921 - 1930
Source: 'Bauhaus prospectus 1929'; as quoted in Artists on Art, from the 14th – 20th centuries, ed. by Robert Goldwater and Marco Treves; Pantheon Books, 1972, London, p. 444

Art, Music, Mathematics, Study

“You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one.”

Jean-Pierre Serre (1926) French mathematician

An Interview with Jean-Pierre Serre - Singapore Mathematical Society https://sms.math.nus.edu.sg/smsmedley/Vol-13-1/An%20interview%20with%20Jean-Pierre%20Serre(CT%20Chong%20&%20YK%20Leong).pdf

“We are so accustomed to hear arithmetic spoken of as one of the three fundamental ingredients in all schemes of instruction, that it seems like inquiring too curiously to ask why this should be. Reading, Writing, and Arithmetic—these three are assumed to be of co-ordinate rank. Are they indeed co-ordinate, and if so on what grounds?
In this modern “trivium” the art of reading is put first. Well, there is no doubt as to its right to the foremost place. For reading is the instrument of all our acquisition. It is indispensable. There is not an hour in our lives in which it does not make a great difference to us whether we can read or not. And the art of Writing, too; that is the instrument of all communication, and it becomes, in one form or other, useful to us every day. But Counting—doing sums,—how often in life does this accomplishment come into exercise? Beyond the simplest additions, and the power to check the items of a bill, the arithmetical knowledge required of any well-informed person in private life is very limited. For all practical purposes, whatever I may have learned at school of fractions, or proportion, or decimals, is, unless I happen to be in business, far less available to me in life than a knowledge, say, of history of my own country, or the elementary truths of physics. The truth is, that regarded as practical arts, reading, writing, and arithmetic have no right to be classed together as co-ordinate elements of education; for the last of these is considerably less useful to the average man or woman not only than the other two, but than 267 many others that might be named. But reading, writing, and such mathematical or logical exercise as may be gained in connection with the manifestation of numbers, have a right to constitute the primary elements of instruction. And I believe that arithmetic, if it deserves the high place that it conventionally holds in our educational system, deserves it mainly on the ground that it is to be treated as a logical exercise. It is the only branch of mathematics which has found its way into primary and early education; other departments of pure science being reserved for what is called higher or university instruction. But all the arguments in favor of teaching algebra and trigonometry to advanced students, apply equally to the teaching of the principles or theory of arithmetic to schoolboys. It is calculated to do for them exactly the same kind of service, to educate one side of their minds, to bring into play one set of faculties which cannot be so severely or properly exercised in any other department of learning. In short, relatively to the needs of a beginner, Arithmetic, as a science, is just as valuable—it is certainly quite as intelligible—as the higher mathematics to a university student.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 267-268.

Life, Truth, Education, School

“Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers… were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians… The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences… Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency… it did trouble deeply the European mathematicians.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p.144

History, Mathematics, Problem, Reading

“The first noteworthy attempt to write an algebra in England was made by Robert Recorde, whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde.
The first Italian textbook to bear the title of algebra was Bombelli's work of 1572.
By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. …this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675).
So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, p. 386, Ch. 6: Algebra,-->

School, Time, Perfection

“My algebra was relatively poor. I found it very difficult to use equations that substituted numbers — to which I had a synesthetic and emotional response — for letters, to which I had none. It was because of this that I decided not to continue math at Advanced level, but chose to study history, French and German instead.”

Daniel Tammet book Born on a Blue Day

Source: Born on a Blue Day (2006), Chapter 6: Adolescence.

History, Study

“I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen’s, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell’s equations.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

"Kenneth E. Iverson" http://keiapl.info/rhui/autobio.htm, autobiographical sketch from an unfinished work (ca. 2004), on his experience at Harvard with "a Masters program in Automatic Data Processing in 1955; in effect, the first computer science program."

Mathematics, Sense, Learning

“There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 6; As cited in: Leandro N. De Castro, Fernando J. Von Zuben, Recent Developments in Biologically Inspired Computing, Idea Group Inc (IGI), 2005 p. 236

Law, Science

“Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.”

René Descartes book La Géométrie

First Book
La Géométrie (1637)

Pleasure, Problem, Science, Training

“In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration.”

George Klir (1932–2016) American computer scientist

Source: An approach to general systems theory (1969), p. 40.

Mathematics

“The university immediately published my pamphlet, and it was sent to ﬁfty People’s Commissariats. It was distributed only in the Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all.
Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all. The number of responses was not very large. There was quite an interesting reference from the People’s Commissariat of Transportation in which some optimization problems directed at decreasing the mileage of wagons was considered, and a good review of the pamphlet appeared in the journal "The Timber Industry."
At the beginning of 1940 I published a purely mathematical version of this work in Doklady Akad. Nauk [76], expressed in terms of functional analysis and algebra. However, I did not even put in it a reference to my published pamphlet—taking into account the circumstances I did not want my practical work to be used outside the country
In the spring of 1939 I gave some more reports—at the Polytechnic Institute and the House of Scientists, but several times met with the objection that the work used mathematical methods, and in the West the mathematical school in economics was an anti-Marxist school and mathematics in economics was a means for apologists of capitalism. This forced me when writing a pamphlet to avoid the term "economic" as much as possible and talk about the organization and planning of production; the role and meaning of the Lagrange multipliers had to be given somewhere in the outskirts of the second appendix and in the semi Aesopian language.”

Leonid Kantorovich (1912–1986) Russian mathematician

L.V. Kantorovich (1996) Descriptive Theory of Sets and Functions. p. 41; As cited in: K. Aardal, ‎George L. Nemhauser, ‎R. Weismantel (2005) Handbooks in Operations Research and Management Science, p. 19-20

War, People, World, School

“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.”

Morris Kline (1908–1992) American mathematician

Morris Kline, p.22.
Mathematics for the Nonmathematician (1967)

History, Mathematics, Idea

“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o' th' day
The clock doth strike, by algebra.”

Samuel Butler (poet) (1612–1680) poet and satirist

Canto I, line 119
Source: Hudibras, Part I (1663–1664)

Mathematics

“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."”

George Peacock (1791–1858) Scottish mathematician

Vol. II: On Symbolical Algebra and its Applications to the Geometry of Position (1845) Ch. XV, p. 59
A Treatise on Algebra (1842)

“At the ULB, Brout and I initiated a research group in fundamental interactions, that is, in the search for the general laws of nature. Joined by brilliant students, many of them becoming world renowned physicists, our group contributed to the many fields at the frontier of the challenges facing contemporary physics. While the mechanism discovered in 1964 was developed all over the world to encode the nature of weak interactions in a "Standard Model," our group contributed to the understanding of strong interactions and quark confinement, general relativity and cosmology. There we introduced the idea of a primordial exponential expansion of the universe, later called inflation, which we related to the origin of the universe itself, a scenario, which I still think may possibly be conceptually the correct one. During these developments, our group extended our contacts with other Belgian universities and got involved in many international collaborations.
With our group and many other collaborators I analysed fractal structures, supergravity, string theory, infinite Kac-Moody algebras and more generally all tentative approaches to what I consider as the most important problem in fundamental interactions: the solution to the conflict between the classical Einsteinian theory of gravitation, namely general relativity, and the framework of our present understanding of the world, quantum theory.”

François Englert (1932) Belgian theoretical physicist

excerpt[François Englert - Biographical, Nobel Prize in Physics (nobelprize.org), 2013, https://www.nobelprize.org/nobel_prizes/physics/laureates/2013/englert-bio.html]

World, Law, Search, Idea

“A colorful hanging chart with no lines.
A pure algebra problem with no solution.”

Shu Ting (1952) Chinese writer

"Missing You" (1978), in Out of the Howling Storm: The New Chinese Poetry, ed. Tony Barnstone (Wesleyan University Press, 1993), p. 61

Color, Problem

“When I had the honour of his conversation, I endeavoured to learn his thoughts upon mathematical subjects, and something historical concerning his inventions, that I had not been before acquainted with. I found, he had read fewer of the modern mathematicians, than one could have expected; but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook. I have often heard him censure the handling geometrical subjects by algebraic calculations; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Des Cartes had given to the treatise, wherein he shews, how the geometer may assist his invention by such kind of computations. He frequently praised Slusius, Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius's book De sectione rationis for giving us a clearer notion of that analysis than we had before.”

Henry Pemberton (1694–1771) British doctor

Preface; The bold passage is subject of the 1809 article " Remarks on a Passage in Castillione's Life' of Sir Isaac Newton http://books.google.com/books?id=BS1WAAAAYAAJ&pg=PA519." By John Winthrop, in: The Philosophical Transactions of the Royal Society of London, from Their Commencement, in 1665, to the Year 1800: 1770-1776: 1770-1776. Charles Hutton et al. eds. (1809) p. 519. Preface to View of Newton's Philosophy, (1728)

Books, Mathematics, Learning, Universe

“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…”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

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Science

“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

Nicholas Murray Butler (1862–1947) American philosopher, diplomat, and educator

Editor's Introduction, The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906) by David Eugene Smith

Mathematics, Study, Universe, Time

“…nor have I found occasion to depart from the plan… the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange… had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The Differential and Integral Calculus (1836)

Idea, Science, Parting

“That a formal science like algebra, the creation of our abstract thought, should thus, in a sense, dictate the laws of its own being, is very remarkable. It has required the experience of centuries for us to realize the full force of this appeal.”

George Ballard Mathews (1861–1922) British mathematician

G.B. Mathews quoted in: F. Spencer. Chapters on Aims and Practice of Teaching, (London, 1899), p. 184. Reported in Moritz (1914).

Law, Sense, Science

“I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

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Mathematics, Question, Learning

“It is less than four years since cohomological methods (i. e. methods of Homological Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper[11], and it seems certain that they are to overflow the part of mathematics in the coming years, from the foundations up to the most advanced parts. … [11] Serre, J. P. Faisceaux algébriques cohérents. Ann. Math. (2), 6, 197–278”

Alexander Grothendieck (1928–2014) French mathematician

1955
[1960, Cambridge University Press, The cohomology theory of abstract algebraic varieties, Proc. Internat. Congress Math.(Edinburgh, 1958), 103–118, https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/CohomologyVarieties.pdf] (p. 103)

Mathematics, Parting

“Men are liars. We'll lie about lying if we have to. I'm an algebra liar. I figure two good lies make a positive.”

Tim Allen (1953) American actor, voiceover artist and comedian

As quoted in Land Your Dream Job : High-Performance Techniques to Get Noticed, Get Hired, and Get Ahead (2007) by John Middleton, Ken Langdon, and Nikki Cartwright

Men, Lie, Lying

“It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B. C. In Liu Hui's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position… and an arrangement of terms in a kind of determinant notation. The rules given by Liu Hui form a kind of rhetorical algebra.
The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving indeterminate equations. …Sun-tzï solved such problems by analysis and was content with a single result…
The Chinese certainly knew how to solve quadratics as early as the 1st century B. C., and rules given even as early as the K'iu-ch'ang Suan-shu… involve the solution of such equations.
Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should…
By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).
The culmination of Chinese is found in the 13th century. …numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner”

David Eugene Smith (1860–1944) American mathematician

1819
Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Way, Question, Problem, Appearance

“It is possible that the major collaboration between economists and engineers is still to come, in the greater use of physical analogues and computers of the analogue type to avoid the difficulties of calculation. Apart from their major use as possible tools for economic regulation, physical analogues have a subsidiary use, for there are students of economics, as there are many students of engineering, who can better understand the significance of the somewhat formidable mathematics that tends to be used in this field, if they can first acquaint themselves with the types of behaviour in question as exhibited by physical objects that can be seen, felt, handled and experimented with. It may also be suggested that economists may find that what they have to say about economic policy will be very readily assimilated by one group of attentive pupils, namely the scientists, engineers and technicians of industry, if explanatory notions can be drawn from the theory of automatic control, which is now part of a normal engineering education. The aim of this essay has been to give explanations of system behaviour, and some approaches to its analysis, using geometrical construction and physical analogy where possible to clarify the implications of the more usual formal algebraic approach.”

Arnold Tustin (1899–1994) British engineer

Source: The Mechanism of Economic Systems (1953), p. 2

Education, Mathematics, Understanding, Question

“The Arithmetica… is deficient, sometimes pardonably, sometimes without excuse, in generalization. The book of Porismata, to which Diophantus sometimes refers, seems on the other hand to have been entirely devoted to the discussion of general properties of numbers. It is three times expressly quoted in the Arithmetica… Of all these propositions he says… 'we find it in the Porisms'; but he cites also a great many similar propositions without expressly referring to the Porisms. These latter citations fall into two classes, the first of which contains mere identities, such as the algebraical equivalents of the theorems in Euclid II…. The other class contains general propositions concerning the resolution of numbers into the sum of two, three or four squares…. It will be seen that all these propositions are of the general form which ought to have been but is not adopted in the Arithmetica. We are therefore led to the conclusion that the Porismata, like the pamphlet on Polygonal Numbers, was a synthetic and not an analytic treatise. It is open, however, to anyone to maintain the contrary, since no proof of any porism is now extant.”

James Gow (scholar) (1854–1923) scholar

p, 125
A Short History of Greek Mathematics (1884)

Books, Quotes, Time

“In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

As quoted in Gauss, Werke, Bd. 8, page 298
As quoted in Memorabilia Mathematica (or The Philomath's Quotation-Book) (1914) by Robert Edouard Moritz, quotation #1215
As quoted in The First Systems of Weighted Differential and Integral Calculus (1980) by Jane Grossman, Michael Grossman, and Robert Katz, page ii

Nature, Problem

David Eugene Smith (1860–1944) American mathematician

David Eugene Smith, "Editor's Introduction," in: The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906)

Mathematics, Study, Universe, Time

“In addition, the teaching of theories from axioms, or some close imitation of them such as the basic laws of an algebra, is usually an educational disaster.”

Ivor Grattan-Guinness (1941–2014) Historian of mathematics and logic

Source: The Rainbow of Mathematics: A History of the Mathematical Sciences (2000), p. 739.

Education, Law

“In the algebra of fantasy, A times B doesn't have to equal B times A. But, once established, the equation must hold throughout the story.”

Lloyd Alexander (1924–2007) American children's writer

"The Flat-Heeled Muse", Horn Book Magazine (1 April 1965)

Time

“Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.”

Antoine Augustin Cournot Researches into the Mathematical Principles of the Theory of Wealth

Source: Researches into the Mathematical Principles of the Theory of Wealth, 1897, p. 4; Cited in: Moritz (1914, 197): About mathematics as language

Pain, Understanding, Reading

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation. p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52 Examples of the processes of the differential and integral calculus, (1841)

Law, Nature, Time, Dependency

“I had almost no education. I hardly went to school. I’d be away for weeks doing films, then I’d come back and have no idea what they were talking about. Of course the other kids loved it: ‘You may be a well-known star but you don’t know how to do algebra.’ It was a nightmare.”

Petula Clark (1932) British actress and singer

Calgary Herald, 14 Feb 2013

Love, Education, School, Star

“The world can no longer accept, the world can no longer accept that basic education is enough. Why do leaders accept that for children in developing countries, only basic literacy is sufficient, when their own children do homework in Algebra, Mathematics, Science and Physics?”

Malala Yousafzai (1997) Pakistani children's education activist

Nobel Peace Prize Lecture (December 10, 2014)

Children, World, Education, Mathematics

“For four or five centuries, Islam was the most brilliant civilization in the Old World. (…) At its higher level the golden age of Muslim civilization was both an immense scientific success and a exceptional revival of ancient philosophy. These was not its only triumphs; literature was another: but they eclipse the rest. First, science: it was there that the Saracens (…) made the most original contributions. These, in brief, were nothing less than trigonometry and algebra (with its significantly Arab name). (…) Equally distinguished were Islam's mathematical geographers, its astronomical observatories and instruments (…). The Muslims also deserve high marks for optics, for chemistry (…) and for pharmacy. More than half the remedies and healing aids used by the West came from Islam (…). Muslim medical skill was incontestable. (…) In the field of philosophy, what took place was rediscovery - a return, essentially of the peripatetic philosophy. The scope of this rediscovery, however, was not limited to copying and handling on, valuable as that undoubtely was. It also involved continuing, elucidating and creating.”

Fernand Braudel (1902–1985) French historian and a leader of the Annales School

A History of Civilizations , Penguin, 1995, p. 73-81

Age, World, Literature, Mathematics

“The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries.”

James Gow (scholar) (1854–1923) scholar

Preface
A Short History of Greek Mathematics (1884)

History, Mathematics

“In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance:—and… the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth—and… the better qualified does he become to free the exposition and application of principles from mathematical intricacy.”

William John Macquorn Rankine (1820–1872) civil engineer

p, 125
"On the Harmony of Theory and Practice in Mechanics" (Jan. 3, 1856)

Truth, Mathematics, Study, Science

“There are many cases of these algebras which may obviously be combined into natural classes, but the consideration of this portion of the subject will be reserved to subsequent researches.”

Benjamin Peirce Linear Associative Algebra

"Natural Classification", p. 119.
Linear Associative Algebra (1882)

Nature

“With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra…”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Science

“Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.”

Fran Lebowitz book Social Studies

"Tips for Teens".
Social Studies (1981)

Life

“There was an NPR story this morning, about the indigenous peoples of Australia, which might make a good column. Apparently they want to preserve their culture, language, and religion because they're slowly disappearing, which is certainly understandable. But, for some reason, they also want more stuff — better education, housing, etc. — from the Australian government. Isn't it odd that it never occurs to such groups that maybe, just maybe, the reason their cultures are evaporating is that they get too much of that stuff already? Indeed, I'm at a loss as to how mastering algebra and biology will make aboriginal kids more likely to believe — oh, I dunno — that hallucinogenic excretions from a frog have spiritual value. And I'm at a loss as to how better clinics and hospitals will do anything but make the shamans and medicine men look more useless. And now that I think about it, that's the point I was trying to get at a few paragraphs ago, when I was talking about the symbiotic relationship between freedom and the hurly-burly of life. Cultures grow on the vine of tradition. These traditions are based on habits necessary for survival, and day-to-day problem solving. Wealth, technology, and medicine have the power to shatter tradition because they solve problems.”

Jonah Goldberg (1969) American political writer and pundit

( August 15, 2001 http://web.archive.org/web/20010105/www.nationalreview.com/goldberg/goldberg081501.shtml) 2000s, 2001

Life, Men, People, Education

“An ounce of algebra is worth a ton of verbal argument.”

J. B. S. Haldane (1892–1964) Geneticist and evolutionary biologist

As quoted in his obituary by Maynard Smith http://www.nature.com/nature/focus/maynardsmith/pdf/1965.pdf in Nature 206 (1965), p. 239

“The second part of the book… contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity.”

E. W. Hobson (1856–1933) British mathematician

A Treatise on Plane Trigonometry https://books.google.com/books?id=_ktLAAAAMAAJ (1891) Preface

Books, Parting

“The absolute scholar is in fact a rather uncanny being. He is instinct with Nietzsche's finding that to be interested in something, to be totally interested in it, is a libidinal thrust more powerful than love or hatred, more tenacious than faith or friendship — not infrequently, indeed, more compelling than personal life itself. Archimedes does not flee from his killers, he does not even turn his head to acknowledge their rush into his garden when he is immersed in the algebra of conic sections.”

George Steiner (1929–2020) American writer

"The Cleric of Treason".
George Steiner: A Reader (1984)

Love, Life, Friendship, Faith

“"Groping" and "muddling through" is usually described as a solution by trial and error…. a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result…. we may wish a better characterization…"successive trials" or "successive corrections" or "successive approximations."… You use successive approximations when… looking for a word in the dictionary… A mathematician may apply the term… to a highly sophisticated procedure… to treat some very advanced problem… that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation… may appear as successive approximations to the truth.
Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for… many… situations, straightforward algebra is more efficient than successive approximations.”

George Pólya (1887–1985) Hungarian mathematician

George Pólya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1962)

Truth, Error, Intelligence, Problem

“I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
A Treatise on Algebra (1842)

Science, Parting

“Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed… all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. vi-vii
A Treatise on Algebra (1842)

“Brook Taylor… in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. …Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 427

Idea, Nature

“Vladimir Voevodsky and his collaborators have provided us with a very interesting candidate-category of motives: a category (of sheaves relative to an extraordinarily fine Grothendieck-style topology on the category of schemes) which in some intuitive sense “softens algebraic geometry” so as to allow for a good notion of homotopy in an algebro-geometric setup and is sufficiently directly connected to concrete algebraic geometry to have already yielded some extraordinary applications. The quest for a full theory of motives is a potent driving force in complex analysis, algebraic geometry, automorphic representation theory, the study of L functions, and arithmetic. It will continue to be so throughout the current century.”

Barry Mazur (1937) American mathematician

"WHAT IS...a Motive?" 2004

Motivation, Sense, Study

“Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 37; Cited in: William Torrey Harris (1879) The Journal of Speculative Philosophy, p. 109

Law, Indifference

“The first writer on algebra whose works have come down to us is Ahmes. He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Problem

“It is vain futility to analyze the algebra of time.”

Dejan Stojanovic book The Creator

“The Day,” p. 57
The Creator (2000), Sequence: “The Whisper of Eternity”

Time

“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.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iv
A Treatise on Algebra (1842)

Science

“I Gladly avail myself of the opportunity of inscribing to you, for a second time, a work of mine on Algebra, as a sincere tribute of my respect, affection and gratitude.
I trust that I shall not be considered as derogating from the higher duties which, (in common with you), I owe to my station in the Church, if I continue to devote some portion of the leisure at my command, to the completion of an extensive Treatise, embracing the more important departments of Analysis, the execution of which I have long contemplated, and which, in its first volume I now offer to the public, under the auspices of one of my best and dearest friends.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra To the Rev. James Tate, M.A. Canon Residentiary of St. Paul's p. i
A Treatise on Algebra (1842)

Time, Sincerity, Gratitude

“The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.”

Niels Henrik Abel (1802–1829) Norwegian mathematician

Hope

“There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; Brahmagupta, whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic… Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)… extends the work through quadratic equations.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Problem

“General systems theory deals with the most fundamental concepts and aspects of systems. Many theories dealing with more specific types of systems (e. g., dynamical systems, automata, control systems, game-theoretic systems, among many others) have been under development for quite some time. General systems theory is concerned with the basic issues common to all these specialized treatments. Also, for truly complex phenomena, such as those found predominantly in the social and biological sciences, the specialized descriptions used in classical theories (which are based on special mathematical structures such as differential or difference equations, numerical or abstract algebras, etc.) do not adequately and properly represent the actual events. Either because of this inadequate match between the events and types of descriptions available or because of the pure lack of knowledge, for many truly complex problems one can give only the most general statements, which are qualitative and too often even only verbal. General systems theory is aimed at providing a description and explanation for such complex phenomena.”

Mihajlo D. Mesarovic (1928) Serbian academic

Mihajlo D. Mesarovic and Y. Takahare (1975) General Systems Theory, Mathematical foundations. Academic Press. Cited in: Franz Pichler, Roberto Moreno Diaz (1993. Computer Aided Systems Theory. p. 134

Game, Mathematics, Knowledge, Problem

“Remember that algebra, with all its deep and intricate problems, is nothing but a development of the four fundamental operations of arithmetic. Everyone who understands the meaning of addition, subtraction, multiplication, and division holds the key to all algebraic problems.”

Richard von Mises (1883–1953) Austrian physicist and mathematician

Second Lecture, The Elements of the Theory of Probability, p. 38
Probability, Statistics And Truth - Second Revised English Edition - (1957)

Understanding, Problem

“The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. …on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries…”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 59

History, Mathematics

“A finished or even a competent reasoner is not the work of nature alone… education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one… in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not.
.. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:—
1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general.
3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion.
4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if… reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded.
…These are the principal grounds on which… the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Source: On the Study and Difficulties of Mathematics (1831), Ch. I.

Life, Children, Truth, Education

“What makes the Nightmares on Elm Street universal and forever in their attraction, I think, is simply the dream. The landscape of the mind, the subconscious, the nightmare. Everybody has a nightmare, and everybody apparently has falling dreams, and everybody has the drowning dream, and everybody has certain kinds of sexual manifestation dreams, as well as our stress dreams; I didn’t study for the algebra test, I didn’t study for my driving test, you know, all those dreams. I still have those dreams, and it’s just such an interesting thing that our mind can turn against us, our own mind, you know we all have.”

Robert Englund (1947) American actor, voice-actor, singer, and director

Robert Englund On El Rey’s 45 Hour ‘A Nightmare on Elm Street’ Marathon, the Passing of Wes Craven, and the Current State of the Slasher Genre http://bloody-disgusting.com/interviews/3379567/3379567/ (February 12, 2016)

Dreams, Thinking, Study, Universe

“You can ask the question about these ancient topics, such as perfect numbers and amicable numbers… 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..”

Carl Pomerance (1944) American mathematician

"Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory" https://www.youtube.com/watch?v=5cU0g9dI1S8&t=9m40s (July, 2013) Erdős Centennial Conference, Budapest.

Question, Problem, Help